How to determine if the function is a one-to-one function and find the formula of the inverse given f(x)=5x^3-7?

Leon Bishop

Leon Bishop

Answered question

2023-03-25

How to determine if the function is a one-to-one function and find the formula of the inverse given f ( x ) = 5 x 3 - 7 ?

Answer & Explanation

srnessgebf

srnessgebf

Beginner2023-03-26Added 8 answers

Let f : , f ( x ) = 5 x 3 - 7 be a function from to .
To demonstrate this, f is injective over and then find its inverse. But we will do it the other way around; ie., we will find the inverse of f and this will be sufficient to prove that f is injective.
f ( x ) = 5 x 3 - 7
so f ( x ) + 7 = 5 x 3
so f ( x ) + 7 5 = x 3
so ( f ( x ) + 7 5 ) 1 3 = x
so f - 1 ( x ) = ( x + 7 5 ) 1 3
Now we check that f ( f - 1 ( x ) ) = x and f - 1 ( f ( x ) ) = x
f ( f - 1 ( x ) ) = 5 ( ( x + 7 5 ) 1 3 ) 3 - 7 = 5 x + 7 5 - 7 = x
f - 1 ( f ( x ) ) = ( 5 x 3 - 7 + 7 5 ) 1 3 = ( 5 x 3 5 ) 1 3 = x
So f - 1 is the inverse of f . Now, since f has an inverse, it must be bijective, and so it must be injective.

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